HAL Id: hal-02077057
https://hal.archives-ouvertes.fr/hal-02077057
Submitted on 22 Mar 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Nested sequents for the logic of conditional belief
Marianna Girlando, Bjoern Lellmann, Nicola Olivetti
To cite this version:
Marianna Girlando, Bjoern Lellmann, Nicola Olivetti. Nested sequents for the logic of conditional
belief. JELIA - European Conference on Logics in Artificial Intelligence, May 2019, Rende, Italy.
�hal-02077057�
belief
?Marianna Girlando1[0000−0002−9384−1356], Bj¨orn Lellmann2[0000−0002−5335−1838],
and Nicola Olivetti3[0000−0001−6254−3754]
1 Aix Marseille Univ, Universit´e de Toulon, CNRS, LIS, Marseille, France and
University of Helsinki, Finland [email protected] 2 Technische Universit¨at Wien, Austria -[email protected] 3
Aix Marseille Univ, Universit´e de Toulon, CNRS, LIS, Marseille, France
Abstract. The logic of conditional belief, called Conditional Doxastic Logic (CDL), was proposed by Board, Baltag and Smets to model revis-able belief and knowledge in a multi-agent setting. We present a proof system for CDL in the form of a nested sequent calculus. To the best of our knowledge, ours is the first internal and standard calculus for this logic. We take as primitive a multi-agent version of the “comparative plausibility operator”, as in Lewis’ counterfactual logic. The calculus is analytic and provides a decision procedure for CDL. As a by-product we also obtain a nested sequent calculus for multi-agent modal logic S5i.
Keywords: Nested sequent calculus · Conditional doxastic logic · Belief revision · Multi-agent epistemic logic
1
Introduction
Knowledge and belief are the most important propositional attitudes to reason about epistemic interaction among agents. Conditional Doxastic Logic (CDL) was proposed by Board [4] and Baltag and Smets [1–3] for modelling both belief and knowledge in a multi-agent setting (see also [14]). Differently from knowledge, the essential feature of beliefs is that they are revisable whenever the agent learns new information. To capture the revisable nature of beliefs, CDL contains the conditional belief operator Beli(C|B), the meaning of which is that agent i would
believe C in case she learnt B. Both unconditional beliefs and knowledge can be defined in CDL : BeliB (agent i believes B) as Beli(B|>), KiB (agent i knows
B) as Beli(⊥|¬B), the latter meaning that i considers impossible (inconsistent)
to learn ¬B. We also consider the comparative plausibility operator A 4i B,
whose reading is that the agent i considers A to be at least as plausible as B. This operator, introduced by Lewis for (single-agent) counterfactual logics is interdefinable with the conditional belief operator; thereby also simple belief and knowledge can be defined directly in terms of it.
?
This work was partially supported by the Project TICAMORE ANR-16-CE91-0002-01 and by WWTF project MA 16-28.
The logic of conditional belief has been significantly employed in game the-ory [17], and it has been used as the basic formalism to study further dynamic extensions of epistemic logics, determined by several kinds of epistemic/doxastic actions. Not surprisingly, the axiomatization of the operator Bel in CDL inter-nalises the well-known AGM postulates of belief revision.
The difference between the conditional belief operator Beli(B|A) and the
simple belief operator Beli(A → B) is illustrated by the following (modified)
example from [17]. Let agent i believe that Jones is a coward, BeliC(j). We
want to express that if the agent is to learn that Jones has been sent to battle, S(j), he would no longer believe that he is a coward (since only brave men are sent to battle). Using the simple belief operator would yield a contradiction, because ¬Beli(S(j) → C(j)) implies ¬BeliC(j). However, if we express it as
¬Beli(C(j)|S(j)), we retain consistency, since ¬BeliC(j) cannot be derived (this
is verified, e.g., using the calculus below). As a second example, consider a variant of the three-wise-men puzzle, where agent i initially believes that she has a white hat, BeliWi. However, if i were to learn that agent j knows the colour of the
hat j herself wears, she would change her beliefs and be convinced that she is wearing a black hat instead: Beli(Bi|Kj(Wj ∧ Bj)). The two formulas are
consistent (assuming ¬(Bi∧ Wi)) as the operator is non-monotonic: Beli(C|A)
does not entail Beli(C|A ∧ B).
The original semantics of CDL is defined in terms of Plausibility Models, i.e., standard epistemic models, where each agent is further equipped with a “com-parative plausibility” relation between worlds used to evaluate her (conditional) beliefs. However, following [8, 9], an alternative semantics is given in terms of multi-agent neighbourhood models, which are essentially a multi-agent version of Lewis’ sphere models for counterfactual logics [10]. In particular, the semantics of CDL coincides with a multi-agent version of Lewis’ logic VTA. Proof-theoretically the logic CDL has not been studied much, the only existing calculus for it being the labelled sequent calculus based on this neighbourhood semantics from [9].
Here we propose the first internal calculus for CDL, meaning that the syntac-tic structures employed in the calculus (nested sequents) have a direct formula translation. Since CDL admits two rather different semantics, the internal calcu-lus presents the advantage of being independent of the choice of the semantics, differently from what happens with a labelled proof system.
Similarly to the calculi for Lewis’ conditional logics in [7], our calculus NCDL
takes as primitive the comparative plausibility operator, albeit in its multi-agent version A 4i B. In order to obtain an internal calculus for CDL, the simple
hypersequent structure used to capture Lewis’ logics in [7], is no longer adequate. To keep track of the “locality” of information for each agent, and to account for beliefs of an agent occurring within the beliefs of another, we use a nested structure, which is not necessary in the single-agent case. The calculus NCDL is
analytic and provides a decision procedure for CDL. Its completeness is proved semantically by extracting a finite countermodel from failed proof search. As mentioned, the epistemic operator Ki is defined in CDL, and it corresponds to
the knowledge operator of multi-agent S5i. Hence, “specialising” the rules of
NCDL to the Ki fragment we obtain a natural nested sequent calculus for S5i.
2
Multi-agent conditional logic CDL
The language of CDL extends propositional logic with operators for (conditional) belief, knowledge, and comparative plausibility, all labelled with an agent. Definition 1. Let A be a set of agents, and let i be an agent. Formulas of CDL are generated as follows, for P propositional variable:
FCDL3 A ::= P | ⊥ | > | A → A | A 4iA | Beli(A|A)
A conditional belief formula Beli(C|B) is read “agent i believes C, given B”.
The meaning of a formula A 4iB is that agent i considers A at least as plausible
as B. The operators of Beli and 4i are interdefinable:
Beli(B|A) ≡ (⊥ 4iA) ∨ ¬((A ∧ ¬B) 4 (A ∧ B))
A 4iB ≡ Beli(⊥|A ∨ B) ∨ ¬Beli(¬A|A ∨ B)
Intuitively, an agent conditionally believes B given A whenever she considers A impossible or she considers A∧¬B to be less plausible than A∧B. Unconditional belief and knowledge can then be defined by these operators as follows4 :
BeliA := Beli(A|>) BeliA := ¬(¬A 4i>) (belief)
KiA := Beli(⊥|¬A) KiA := ⊥ 4i ¬A (knowledge)
An axiomatization of CDL is given by the following axioms and rules [4, 3]: (0) Axiomatization of classical propositional logic
(1) If ` B, then ` Beli(B|A)
(2) If ` A ↔ B, then ` Beli(C|A) ↔ Beli(C|B)
(3) (Beli(B|A) ∧ Beli(B → C|A)) → Beli(C|A)
(4) Beli(A|A)
(5) Beli(B|A) → (Beli(C|A ∧ B) ↔ Beli(C|A))
(6) ¬Beli(¬B|A) → (Beli(C|A ∧ B) ↔ Beli(B → C|A))
(7) Beli(B|A) → Beli(Beli(B|A)|C)
(8) ¬Beli(B|A) → Beli(¬Beli(B|A)|C)
(9)A → ¬Beli(⊥|A)
These axioms represent an “internalised” version of the AGM belief revision postulates in a multi-agent setting, e.g., axioms 5 and 6 encode the Minimal Change Principle5 An alternative axiomatization of CDL taking 4i as primitive
4 An equivalent definition of the simple belief operator is the following: Bel iA :=
¬(¬A 4iA) [10]. We choose a simpler formulation in terms of >, also from [10]. 5
essentially amounts to a multi-agent version of Lewis’ counterfactual logic system VTA [7] and contains in addition to classical propositional logic the following:
(CPR) ` B → A ` A 4iB (CPA) (A 4iA ∨ B) ∨ (B 4i A ∨ B) (TR) (A 4i B) ∧ (B 4iC) → (A 4i C) (CO) (A 4iB) ∨ (B 4iA) (N) ¬(⊥ 4i>) (T) (⊥ 4i¬A) → A (A1) (A 4iB) → ⊥ 4i ¬(A 4iB) (A2) ¬(A 4iB) → ⊥ 4i(A 4iB)
The original semantics of CDL is given in terms of plausibility models; the alter-native semantics in terms of neighbourhood models from [9] is as follows. Definition 2. Let A be a set of agents; a multi-agent neighbourhood model has the form M = hW, {Ni}i∈A,J Ki where W is a non empty set of worlds, J K : Atm → P (W ) is the evaluation for atomic formulas and for each i ∈ A, Ni: W → P(P(W )) is a neighbourhood function, satisfying:
– Non-emptiness: For all α ∈ Ni(x), α 6= ∅
– Nesting: For all α, β ∈ Ni(x), α ⊆ β or β ⊆ α
– Total reflexivity: There exists α ∈ Ni(x) such that x ∈ α
– Local absoluteness: If α ∈ Ni(x) and y ∈ α then Ni(x) = Ni(y)
The truth conditions for Boolean combinations of formulas are standard; the remaining ones use the local forcing notation introduced in [12], i.e., α ∀A iff
for all y ∈ α we have y A, and α ∃A iff there exists y ∈ α such that y A: x Beli(B|A) iff for all α ∈ Ni(x) it holds that α ∀¬A or there exists
β ∈ Ni(x) such that β ∃A and β ∀A → B
x A 4iB iff for all β ∈ Ni(x) if β ∃B then β ∃A
x BeliB iff there exists β ∈ Ni(x) such that β ∀B
x KiB iff for all β ∈ Ni(x) it holds that β ∀B
A formula A is valid in M if for all w ∈ W , w A. A formula A is valid if A is valid in every multi-agent neighbourhood model.
3
Nested sequent calculus N
CDLIn this section we present a sequent for CDL. The calculus is based on the structure of nested sequents (e.g., [5, 16]), adjusted to the multiagent setting and extended with the mechanism to handle comparative plausibility formulas using conditional blocks from [6, 13] as follows.
Definition 3. A multi-agent conditional block for agent i is a syntactic struc-ture i: (A1. . . An Ci B), interpreted as: (A1∨ · · · ∨ An) 4i B. A multi-agent
nested sequent (short: nested sequent) S is a structure S = Γ ⇒ ∆, [G1]i1, . . . , [Gn]in
where i1, . . . , in ∈ A, Γ is a multiset of formulas, and ∆ is a multiset of formulas
Intuitively, a nested sequent is a finite labelled directed tree with nodes labelled with sequents Γ ⇒ ∆, where ∆ also contains multi-agent conditional blocks, and edges labelled with agents. We call the nodes with their sequent label the components of the nested sequent. Thus each Gjrepresents an immediate subtree
of the tree with root S. The formula interpretation is given by: (Γ ⇒ ∆, (Σ1CiC1), . . . , (ΣkCjCk), [G1]i1, . . . , [Gn]in)int:= ^ Γ →_∆ ∨_ 16s6k(( _ Σs) 4i Cs) ∨ Ki1(G1) int∨ · · · ∨ K in(Gn) int
for KiA = ⊥ 4i¬A. We sometimes include nested successors into the succedent
of a sequent, denoted with superscript ∗. E.g., for Γ ⇒ ∆, [G]i we also write Γ ⇒ ∆∗. For a multiset ∆, we write set(∆) for its underlying set, i.e., its carrier. To operate with nested sequents, we use the notion of context, denoting a nested sequent with a unique “hole”, to be filled with another nested sequent. Definition 4. We define a context G{ } as:
– G{ } = Γ ⇒ ∆∗, { } is a context;
– if F { } is a context, then G{ } = Γ ⇒ ∆∗, [F { }]i is a context.
The result of filling a context G{ } with a nested sequent Γ ⇒ ∆∗then is denoted as G{Γ ⇒ ∆∗} and defined via:
– If G{ } = Σ ⇒ Π∗, { }, then G{Γ ⇒ ∆∗} = Γ, Σ ⇒ ∆∗, Π∗;
– If G{ } = Σ ⇒ Π∗, [F { }]i then G{Γ ⇒ ∆∗} = Σ ⇒ Π∗, [F {Γ ⇒ ∆∗}]i.
The rules of the multi-agent nested sequent calculus NCDL are given in Fig. 1.
They are formulated in the cumulative version, repeating all formulas and blocks of the conclusion in the premisses. This is used for proving completeness, but of course could be avoided at the cost of explicit contraction rules.
As in nested calculi, each nested sequent can be thought as encoding the formulas relative to one world of the model. Since our neighbourhood models are multi-agent, each nested sequent has associated a label for an agent.
More in detail, rule R 4 introduces backwards a conditional block, and rule L 4 (read upwards) combines a the true plausibility formula in the antecedent of a sequent with a with the false conditional block in the consequent by means of a case analysis. With the com rule, two blocks communicate with one another. This rule can be thought as a syntactic equivalent of the nesting condition over neighbourhoods, with each conditional block encoding the comparative plausibil-ity formulas relative to one neighbourhood of the model. The jump rule creates a new nested sequent in correspondence to a conditional block, with the same agent label. Rule T accounts for the condition of total reflexivity of the neighbourhood function, and the transfer rules are needed to express local absoluteness: due to this condition comparative plausibility formulas are evaluated in the same way at all the worlds accessible for the same agent - and thus, these formulas are allowed to “pass” between nested sequents with the same agent label. Finally, the rules of conditional belief make use of the definition of Beli(B|A) in terms of
the comparative plausibility operator given in the previous section. For instance, rule BelL read backwards states that if Beli(B|A) is true, either A is impossible
(left premiss) or A ∧ ¬B is strictly less plausible than A (right premiss).
Initial sequents G{p, Γ ⇒ ∆, p} G{⊥, Γ ⇒ ∆} Propositional rules G{A → B, B, Γ ⇒ ∆} G{A → B, Γ ⇒ ∆, A} G{A → B, Γ ⇒ ∆} L → G{A, Γ ⇒ ∆, A → B, B} G{Γ ⇒ ∆, A → B} R → Rules for comparative plausibility and conditional blocks
G{Γ ⇒ ∆, A 4iB, (A CiB)}
G{Γ ⇒ ∆, A 4iB} R 4
G{A 4iB, Γ ⇒ ∆, (Σ CiC), (Σ CiA)} G{A 4iB, Γ ⇒ ∆, (B, Σ CiC), (Σ CiC)}
G{A 4iB, Γ ⇒ ∆, (Σ CiC)} L 4
G{Γ ⇒ ∆, (Σ CiC), [C ⇒ Σ]i}
G{Γ ⇒ ∆, (Σ CiC)}
jump
{Γ ⇒ ∆, (Σ1, Σ2CiA), (Σ1CiA), (Σ2CiB)} G{Γ ⇒ ∆, (Σ1CiA), (Σ2CiB), (Σ1, Σ2CiB)}
G{Γ ⇒ ∆, (Σ1CiA), (Σ2CiB)}
com
G{A 4iB, Γ ⇒ ∆, (⊥ CiA)} G{A 4iB, Γ ⇒ ∆, B}
G{A 4iB, Γ ⇒ ∆} T Transfer rules G{A 4iB, Γ ⇒ ∆, [A 4iB, Σ ⇒ Π]i} G{A 4iB, Γ ⇒ ∆, [Σ ⇒ Π]i} Tr1 G{A 4iB, Γ ⇒ ∆, [A 4iB, Σ ⇒ Π]i} G{Γ ⇒ ∆, [A 4iB, Σ ⇒ Π]i} Tr2 G{Γ ⇒ ∆, (Λ CiC), [Σ ⇒ Π, (Λ CiC)]i} G{Γ ⇒ ∆, (Λ CiC), [Σ ⇒ Π]i} Tr3 G{Γ ⇒ ∆, (Λ CiC), [Σ ⇒ Π, (Λ CiC)]i} G{Γ ⇒ ∆, [Σ ⇒ Π, (Λ CiC)]i} Tr4
Rules for conditional belief
G{(A ∧ ¬B) 4iA, Γ ⇒ ∆, (⊥ CiA)}
G{Γ ⇒ ∆, Beli(B|A)}
BelR
G{⊥ 4iA, Beli(B|A), Γ ⇒ ∆} G{Beli(B|A), Γ ⇒ ∆, (A ∧ ¬B CiA)}
G{Beli(B|A), Γ ⇒ ∆}
BelL
Fig. 1. Nested calculus NCDL
Theorem 1 (Soundness). If G is derivable in NCDL then (G)int is valid.
Proof. By induction on the derivation height, showing that if the premiss of a rule is valid, so is its conclusion. By means of example we show jump, T and Tr1.
Suppose the premiss of jump is valid, and its conclusion is not. Thus, there exists a model such that M, x F for all F ∈ Γ and M, x 1 H for all H ∈ ∆.
Since x 1 (Σ C C), there exists α ∈ Ni(x) such that α ∃ C and α 1∃
(A1∨ · · · ∨ An), for Σ = A1, . . . , An. Then there exists y ∈ α such that M, y C
and M, y 1 (A1∨ · · · ∨ An). However, from the previous conditions and validity
of the premiss we have that for all k ∈S Ni(x) either M, k 1 C or M, k As,
As for T, suppose the premisses of the rule are valid, while the conclusion is not. Thus, there is a model M, x A 4i B, such that for all F ∈ Γ , H ∈ ∆,
M, x F and M, x 1 H. From M, x A 4i B we have for all α ∈ Ni(x), if
α ∃B, then α ∃ A. As for the premisses, it must hold that M, x ⊥ 4i A,
and thus that (∗) for all α ∈ Ni(x), α 6 ∃A and M, x B. By total reflexivity,
there is a β ∈ Ni(x) such that x ∈ β. Thus, β ∃ B, whence β ∃ A, which
contradicts (∗).
Similarly, suppose the premiss of Tr1 is valid, while the conclusion is not.
Then there is a model such that M, x A 4i B, and for all F ∈ Γ , H ∈ ∆,
M, x F and M, x 1 H. Moreover, we have that there exists y ∈S Ni(x) such
that M, y S for all S ∈ Σ, and M, y 1 P , for all P ∈ Π. From all these conditions, and from the fact that the premiss of Tr1 are assumed to be valid,
we obtain in particular that (?) M, y 1 A 4i B. However, by local absoluteness
we have Ni(x) = Ni(y); thus M, x A 4iB, against (?). ut
Lemma 1. The rules of weakening and contraction are admissible in NCDL:
G{Γ ⇒ ∆∗} G{Γ, Σ ⇒ ∆∗, Π∗} W G{Γ, A, A ⇒ ∆∗} G{Γ, A ⇒ ∆∗} CL G{Γ ⇒ ∆∗, A, A} G{Γ ⇒ ∆∗, A} CR
Proof. Standard, by induction on the depth of the derivation. ut
Remark 1. The rules for simple belief and knowledge can be explicitly defined as follows: G{Γ ⇒ ∆, (¬A Ci>)} G{Beli A, Γ ⇒ ∆} BL G{¬A 4i>, Γ ⇒ ∆} G{Γ ⇒ ∆, BeliA} BR G{KiA, Γ ⇒ ∆, (¬A, Σ CiC)} G{KiA, Γ ⇒ ∆, (Σ CiC)} KL G{Γ ⇒ ∆, (⊥ CiA)} G{Γ ⇒ ∆, KiA} KR
Example 1. A derivation of KiA → Beli (¬Belj(⊥|A)) is shown in Fig. 2, with
rule R¬ derivable from R →, recalling ¬A = A → ⊥. We omit repetitions of the principal formulas in the premisses.
4
Completeness of N
CDLTo prove completeness of NCDL, we show how to construct a countermodel from
failed proof-search. For this, we first introduce the notion of saturated sequent (Definition 7), i.e., an unprovable sequent to which all the rules have been non-redundantly applied. Then, we build a countermodel for the sequent placed at the root of the derivation from the information contained in the saturated sequent. Intuitively, we can consider a saturated sequent S as a labelled tree, where each node is a nested component Sj of S. Each world of a countermodel for S
corresponds to a node of the tree, and the world falsifying S as a whole is the node placed at the root of the tree.
In countermodel construction we have to take care of the following: (a) for each agent i and world Sj define a system of neighbourhoods Ni(Sj); and (b)
Belj(⊥|A) 4iKiA ⇒ [⊥ 4iA, Belj(⊥|A), A ⇒ ⊥, A]i
Belj(⊥|A) 4iKiA ⇒ [⊥ 4iA, Belj(⊥|A) ⇒ ¬A, ⊥, A]i
R¬ (2) Belj(⊥|A) 4iKiA ⇒ [⊥ 4iA, Belj(⊥|A) ⇒ ¬A, ⊥]i
T (1) Belj(⊥|A) 4iKiA ⇒ [Belj(⊥|A) ⇒ ¬A, ⊥]i
BelL
Belj(⊥|A) 4iKiA ⇒ (¬A, ⊥ CiBelj(⊥|A))
jump Belj(⊥|A) 4iKiA ⇒ (⊥ CiBelj(⊥|A))
KL
· · · ⇒ > Belj(⊥|A) 4iKiA ⇒
T KiA ⇒ Beli(¬Belj(⊥|A))
BR
⇒ KiA → Beli(¬Belj(⊥|A))
R → The derivations of sequents (1) and (2) respectively are:
· · · ⇒ [· · · ⇒ ¬A, ⊥, [A ⇒ A]j
]i · · · ⇒ [· · · ⇒ ¬A, ⊥, [A ⇒ >]j
]i Belj(⊥|A) 4iKiA ⇒ [Belj(⊥|A) ⇒ ¬A, ⊥, [A ⇒ A ∧ >]j]i
R∧ Belj(⊥|A) 4iKiA ⇒ [Belj(⊥|A) ⇒ ¬A, ⊥, (A ∧ > CjA)]i
jump
Belj(⊥|A) 4iKiA ⇒ [Belj(⊥|A) ⇒ ¬A, ⊥, [⊥ ⇒ ⊥]j]i
Belj(⊥|A) 4iKiA ⇒ [Belj(⊥|A) ⇒ ¬A, ⊥, (⊥ Cj⊥)]i
jump
Fig. 2. Derivation of the formula KiA → Beli(¬Belj(⊥|A)).
verify that the condition of local absoluteness holds in the model. Concerning (a), the neighbourhoods Ni(Sj) will be determined by the blocks (Σ Ci C)
contained in the consequent of Sj. As for (b), we need our models to satisfy
the following property. Let M be an arbitrary model, x, y two worlds in the model, and Ri(x, y) the relation defined as y ∈S Ni(x). By local absoluteness
it follows that Ri is an equivalence relation6 and from Ri(x, y) follows Ni(y) =
Ni(x). The syntactic counterpart of Ri is the equivalence relation ∼i between
two components Sj and Sk of S, one of which might be S itself (Definition
6). This relation holds whenever Sj and Sk are related by an i-path in the tree
associated with S. Lemma 2 proves that if Sj ∼iSkthen the two nested sequents
contain the same blocks. This suffices to ensure that Ni(Sj) = Ni(Sk).
Let us come back to (a). To define the set Ni(Sj) for a world Sj, we consider
the blocks (Σ CiC) occurring in the consequent of Sj. However since the rules
are cumulative, Sj may contain two blocks (A1, A2 Ci C) and (A1, A2, A3 Ci
C). In this case the former block can be disregarded, as it is included in the latter. Thus, only “maximal” blocks (Definition 8) are relevant in order to define Ni(Sj). It turns out that maximal blocks of a saturated sequent are ordered
by set inclusion, due to the com rule. Moreover, each maximal block (Σ Ci
C) occurring in Sj is supposed to be false in world Sj. This means that Sj
has associated a “witnessing” world Sk where C is true and all formulas in
6
Σ are false. This world / component is such that Sj ∼i Sk, and its existence
is guaranteed by saturation with respect to jump. Thus, the neighbourhoods Ni(Sj) are determined by the maximal blocks and their witnessing worlds. The
following example should illustrate the construction. Example 2. For pi, r, s, t, u, distinct atomic formulas, let:
Π = (p1Cir), (p1Cis), (p1, p2Cit), (p1, p2, p3Ciu)
S = c ⇒ Π, [r ⇒ p1, Π]i, [s ⇒ p1, Π]i, [t ⇒ p1, p2, Π]i, [u ⇒ p1, p2, p3, Π]i
The four components of S are numbered as S1, S2, S3, S4 respectively (so that
S1= [r ⇒ p1, Π]i etc.). Sequent S is saturated according to Definition 7.
More-over, observe that the blocks in Π are ordered by set inclusion, and that each block has an associated witnessing world: (p1Cir) is associated to S1, (p1Cis)
to S2, (p1, p2 Ci t) to S3 and (p1, p2, p3 Ci u) to S4. In the countermodel,
W = {S, S1, S2, S3, S4}. The system of neighbourhoods Ni(S) is determined by
putting in the smallest neighbourhood the worlds corresponding to the largest block, and so on.
Ni(S) = {{S4}, {S4, S3}, {S4, S3, S2, S1}, {S4, S3, S2, S1, S}}
This ensures that if a neighbourhood α falsifies a block (Σ Ci C), i.e., α ∃C
and α 6 ∃ W Σ, then any larger neighbourhood falsifies the block as well. The inclusion of S in the largest sphere is needed to ensure total reflexivity. Since the worlds are related by ∼i, we have that Ni(Sj) = Ni(S). Finally, the evaluation
function assigns to atoms the worlds / nested component containing the atoms in the antecedent. Thus,JuK = {S4},JtK = {S3},JuK = {S2},JrK = {S1},JcK = {S }. It can be easily seen that world S falsifies sequent S : for instance, in case of block (p1, p2Cit), we have {S4, S3} ∃t but {S4, S3} 6 ∃p1∨ p2.
Definition 5. Let S1 and S2 be two nested sequents. We say that S2 occurs in
S1, in symbols S2∈ S˜ 1 if S1= S2or S1= Γ ⇒ ∆∗, [S3]i for some i and S2∈ S˜ 3.
Viewing nested sequents as labelled trees, we thus have S2∈ S˜ 1if S2is a subtree
of S1. We denote by the symbol ∈ occurrence of a formula A or conditional block
(Σ CiA) in a multiset Γ of formulas and conditional blocks.
Definition 6. Let S be a nested sequent. For every agent i the relation ∼i on
the nested sequents occurring in S is the equivalence relation generated by the relation ∼1i given by: S1∼1i S2 iff S1= Γ ⇒ ∆, [S2]i.
Intuitively, we have S1∼i S2 if S1= S2 or the two components are linked with
an i-path. Next, recall that set(∆) is the set underlying the multiset ∆.
Definition 7. Let S = Γ ⇒ ∆, [G1]i1, . . . [Gn]in be a nested sequent. We say
that S is locally saturated if it satisfies the following conditions. 1. (init) Γ ∩ ∆ = ∅ and ⊥ /∈ Γ ;
2. (L →) If A → B ∈ Γ then A ∈ ∆ or B ∈ Γ ; 3. (R →) If A → B ∈ ∆ then A ∈ Γ and B ∈ ∆;
4. (R 4) If A 4iB ∈ ∆ then there exists a conditional block (A Ci B) ∈ ∆;
5. (L 4) If A 4iB ∈ Γ and (Σ CiC) ∈ ∆, then there is a (Σ0 Ci C) ∈ ∆ with
set(Σ, B) = set(Σ0) or (Σ CiA) ∈ ∆;
6. (com) If (Σ1Ci A) and (Σ2CiB) ∈ ∆, then for some Π with set(Σ1, Σ2) ⊆
set(Π) we have (Π CiA) ∈ ∆ or (Π Ci B) ∈ ∆.
7. (T) If A 4iB ∈ ∆ then either (⊥ CiA) ∈ ∆ or B ∈ ∆;
We denote by Blocki(S) the set of conditional blocks in ∆ labelled with i.
More-over, we say that S is saturated if the following conditions hold for every S1∈ S:˜
– S1 is locally saturated;
– (jump) If S1= Γ ⇒ ∆∗, (Σ Ci C), then there is a S2∼iS1 with S2= Φ ⇒
Ω∗, [Ψ, C ⇒ Σ, Ξ∗]i;
– (Transfer rules) If S1= Γ ⇒ ∆∗, [Σ ⇒ Π∗]i, then Blocki(S1) = Blocki(Σ ⇒
Π∗) and for every formula A 4
iB we have A 4iB ∈ Γ iff A 4i B ∈ Σ;
Lemma 2. If S1and S2are saturated and S1∼i S2, then Blocki(S1) = Blocki(S2).
Proof. By induction on the length of the i-path between S1 and S2, using the
saturation condition for the transfer rules in the base case. ut
We define a naive backwards proof-search strategy for NCDLas follows: Apply the
rules bottom-up to the nested sequent unless the saturation condition associated to the particular application of the rule is already satisfied. If the sequent is saturated and not an initial sequent, return it, otherwise return “derivable”. Lemma 3. Let S be a nested sequent. Then proof search under the strategy above terminates and yields a derivation or a saturated nested sequent.
Proof. For termination, we first bound the number of the nested sequents oc-curring in the proof search. Let n be the size of S, i.e., the number of symbols occurring in it. Note that the premisses of the rules contain at least one for-mula occurrence more than the conclusion. Since according to the proof-search strategy rules are not applied if the nested sequent already satisfies the cor-responding saturation condition, no formula or block is added twice. Since S contains at most n many formulas, at most 2n· n many different conditional blocks and 2n· 2n many sequents consisting only of formulas can be obtained
without repetition. Hence at most 22n·n· 22n many different sequents consisting
of formulas and blocks occur in the proof search. To bound the maximal depth of a nested sequent (seen as a tree) occurring in the proof search, we consider a branch in such a nested sequent and divide it into blocks, taking two components S1 and S2 in the branch to be in the same block if for some agent i we have
S1∼iS2. Since the maximal nesting depth of comparative plausibility formulas
in S is n, the number of alternations between agents in such a formula is at most n. Every application of the jump rule produces a new component such that the maximal nesting depth of formulas in this component is strictly smaller than
that of the component from which it was created. Moreover the transfer rules only transfer comparative plausibility formulas and blocks across nesting oper-ators for the same agent. Hence every branch of every nested sequent occurring in the proof search contains at most n many non-trivial blocks in addition to those of S. Thus the maximal depth of a nested sequent occurring in the proof search is the number of possible sequents times the maximal number of blocks in a branch, i.e., 22n·n· 22n· 2n = O(22n). Since the branching of the nested
sequents themselves (seen as trees) is caused by applications of the jump rule, by the saturation conditions the branching of a nested sequent is bounded by the number of formula-formula sequents, i.e., 22n. Hence the number of components of a nested sequent occurring in the proof search is O((22n)22n). Further, each of these components contains one of at most O(22n) many sequents. Hence the total number of nested sequents which might occur in the proof search is finite. Together with the fact that in every step of the proof search at least one new occurrence of a formula is added, this means that the algorithm terminates.
It is straightforward to construct a derivation if the procedure returns “deriv-able”. Suppose that it does not yield a derivation. Since the algorithm terminates, it yields a nested sequent S. But this nested sequent must satisfy the saturation conditions for every rule, since otherwise it would be possible to apply the
cor-responding rule and the procedure would not have terminated. ut
We then construct a countermodel from a saturated nested sequent. While the worlds of the model will be the components of the nested sequents, for defining the neighbourhood function we consider the “largest” blocks in the components: Definition 8. For a nested sequent S, a conditional block (Σ CiC) ∈ Blocki(S)
is maximal if there is no block (Σ0 Ci C) ∈ Blocki(S) with set(Σ) ( set(Σ0).
We write MaxBlocki(S) for the set of maximal blocks in Blocki(S).
Remark 2. A maximal conditional block is the “largest” (containing most for-mulas in the antecedent) of all the blocks in Blocki(S) with the same consequent.
Thus, all maximal blocks have a different consequent. If S is saturated, the an-tecedents of the conditional blocks in MaxBlocki(S) can be ordered w.r.t. set
inclusion, such that set(Σ1) ⊂ set(Σ2) ⊂ · · · ⊂ set(Σk), for k the number of
maximal conditional blocks. Note that there could be maximal blocks sharing the same antecedent, e.g., as a consequence of saturation with respect to Tr3,
Tr4or com, this latter applied to two different pairs of conditional blocks.
Given a saturated nested sequent S as above, the construction of the counter-model MN = hW, {Ni}i∈A,J Ki proceeds as follows.
– W := {Sj | Sj∈ S};˜
– JpK := {Sj∈ W | p ∈ Φj}.
To define the neighbourhood functions, observe that by the condition of abso-luteness, this must be the same for all worlds seen by the same agent. Thus, for all nested sequents Smwith Sm∼iSj, we define a single neighbourhood function
in Sj, knowing by Lemma 2 that if Sj ∼iSmthen Blocki(Sj) = Blocki(Sm), and
hence MaxBlocki(Sj) = MaxBlocki(Sm). Suppose the set MaxBlocki(Sj)
con-tains n1+ n2+ · · · + nkmaximal conditional blocks, with exactly k different sets
set(Σ1) ⊂ set(Σ2) ⊂ · · · ⊂ set(Σk):
(Σ1CiC11) , . . . , (Σ1CiCn11)
(Σ2CiC12) , . . . , (Σ2CiCn22)
..
. ...
(ΣkCiC1k) , . . . , (Σk CiCnkk)
So for each z 6 k there are nzdifferent blocks (ΣzCiC1z), . . . , (ΣzCiCnzz) with
the same antecedent. By the saturation condition for jump, for all Σz, Cwz with
w ∈ {1, . . . , nz}, there is a Sz,w = Φz,w⇒ Ωz,w∈ S with S˜ j∼iSz,w, Cwz ∈ Φz,w
and Σz⊆ Ωz,w. Let W Sj
i = {Sz| Sz∼iSj}. Now define Ni(Sj) as follows:
Ni(Sj) := {{Sk,1, . . . , Sk,nk}, {Sk,1, . . . , Sk,nk, Sk−1,1, . . . , Sk−1,nk−1}, . . . ,
{Sk,1, . . . , Sk,nk, Sk−1,1, . . . , Sk−1,nk−1, . . . S1,1, . . . , S1,n1}, W
Sj
i }
I.e., we add into the same neighbourhood the worlds associated to blocks sharing the same antecedent. The so defined MN is a model for CDL: it satisfies the
properties of non-emptiness, nesting and local absoluteness (immediate from the definition). Total reflexivity follows from the fact that for all Sj, W
Sj
i ∈ Ni(Sj).
Lemma 4. Let S be a saturated nested sequent and Sj= Φj⇒ Ωj∗ a nested
se-quent with Sj ∼iS. Let MN be the model as just defined. Let MaxBlocki(Sj) =
(Σ1 Ci C11), . . . , (Σ1 Ci Cn11), . . . , (Σk Ci C
k
1), . . . , (Σk Ci Cnkk). For A a
for-mula and (Σ CiC) a conditional block the following hold:
1. If A ∈ Φj then MN, Sj A;
2. If A ∈ Ωj∗ then MN, Sj1 A;
3. If (Σ CiC) ∈ Ωj∗ then MN, Sj 1 (WB∈ΣB 4iC).
Proof. We prove statements 1 and 2 by induction on the complexity of A, show-ing only the case of comparative plausibility formulas. The proof of statement 3 uses the proof of 2. As for 1, suppose A 4i B ∈ Φj. We have to show that
MN, Sj A 4i B, i.e. that for all the α ∈ Ni(Sj) we have α 1∃B or α ∃ A.
First, suppose α 6= WSj
i . Then, α = {Sk,1, . . . , Sk,nk, . . . , St,1, . . . , St,nt}, for
some t 6 k. For z 6 k and w ∈ {1, . . . , nz}, each Sz,w comes from a maximal
conditional block (Σz Ci Cwz), and denotes a nested sequent Φz,w ⇒ Ωz,w
oc-curring in W with Cz
w ∈ Φz,w and Σz ⊆ Ωz,w. By saturation condition L 4,
either B ∈ Σt or A = Cqt, for some q ∈ {t, . . . , nt}. In the former case, by
set(Σt) ⊂ set(Σt+1) ⊂ · · · ⊂ set(Σk) and by inductive hypothesis, we have that
for all Sz,w, with z 6 k and w ∈ {1, . . . , nz}, MN, Sz,w 1 B; thus, α 1∃ B. Otherwise, let A = Ct
q, for some q ∈ {1, . . . , nt}. Then, St,q = A, Φ0t,q ⇒ Ωt,q.
By inductive hypothesis and since St,q∈ α we get α ∃A.
If α = WSj
i , we have to prove that W
Sj i 1∃ B or W Sj i ∃ A. Let W Sj i =
for all q 6 t. By saturation condition T, either there exists some Sq with (⊥ Ci
A) ∈ Blocki(Sq), or for all Sq we have B ∈ Ωq. In the former case, by saturation
condition jump, to Sq is associated a nested sequent Sq0 = A, Φq0 ⇒ Ωq0. It holds
that Sq ∼1i Sq0, and thus Sq0 ∈ WiSj. By inductive hypothesis, M, Sq0 A, and
WSj
i ∃A. Otherwise, we have that for all Sq, B ∈ Ωq. By inductive hypothesis
M, Sq 1 B, and thus WiSj 1 ∃B.
As for 2, suppose A 4i B ∈ Ωj. We have to prove that MN, Sj 1 A 4 B, i.e., that there is an α ∈ Ni(Sj) with α ∃B and α ∃B and α 1∃A. From the
definition of Ni(Sj), and with z 6 k and w ∈ {1, . . . , nz}, we have that to each
Sz,w occurring inS Ni(Sj) is associated a sequent Cwz, Φz,w ⇒ Ωz,w, Σz, coming
from a maximal conditional block (Σz Ci Cwz). Thus, by saturation for R 4
there exists z 6 k and w ∈ {1, . . . , nz} such that B = Cwz and A ∈ Σz. Let us
consider the world Sz,wassociated to this nested sequent, and the sphere to which
Sz,w belongs: α = {Sk,1, . . . Sk,nk. . . , Sz,1, . . . , Sz,nz}. By inductive hypothesis,
MN B, and thus α ∃ B. Moreover, since set(Σz) ⊂ set(Σz+1) ⊂ · · · ⊂
set(Σk) and by inductive hypothesis, it holds that for all Sl,q, for l ∈ {z, . . . , k}
and q ∈ {1, . . . , nl}, Sl,q1 A. Since no worlds in α validate A, α 1∃A. ut Corollary 1. Let S = Γ ⇒ ∆, [G1]i1, . . . , [Gn]in be a saturated nested sequent
and MN a model as defined above. Then, for all Sj ∈ W it holds that MN, Sj1
(Sj)int, and MN, S 1 (S)int. ut
Completeness of NCDL follows immediately: by Lem. 3, backwards proof search
terminates, yielding a derivation or a saturated sequent. In the former case the formula is derivable; in the latter case, we obtain a countermodel using Cor. 1. Moreover, the completeness proof constructs a finite countermodel from a satu-rated sequent, and thereby also shows the finite model property of the logic.
Theorem 2 (Completeness). Every valid formula is derivable in NCDL. ut
Example 3. We construct the countermodel M for the underivable sequent ⇒ Beli(P → Q) → Beli(Q|P ). By backwards applications of NCDL rules we obtain
the following saturated sequent, where we assume 4i binds stronger than ∧:
S = P ∧ ¬Q 4iP, Blocki(S) ⇒ [P ⇒ Q, Blocki(S)]i, [> ⇒ P, Blocki(S)]i
where Blocki(S) = (⊥ Ci P ∧ ¬Q), (P ∧ ¬Q, P, ⊥ Ci >). Let S1 = P ⇒
Q, Blocki(S) and S2 = > ⇒ P, Blocki(S). Then, W = WiS = {S, S1, S2}, and
S ∼i S1 ∼i S2. Sequent S1 and S2 are obtained by jump respectively from the
former and latter conditional block in Blocki(S). Since {⊥} ⊂ {P ∧ ¬Q, P, ⊥}
we have that Ni(S) = Ni(S1) = Ni(S2) = {{S2}, {S2, S1}, WiS}. By
defini-tion, P is true only at world S2, and Q is false at all the worlds. It holds
that i) M, S Beli(P → Q), i.e., that there exists an α ∈ Ni(S) such that
α ∀ P → Q. Neighbourhood {S2} satisfies the condition. It also holds that
ii) M, S 1 Beli(Q|P ), i.e., that there exists an α ∈ Ni(S) such that α ∃ P
and that for all β ∈ Ni(S) it holds that β ∃ P ∧ ¬Q. The former condition
is satisfied by the neighbourhood {S2, S1}, and all neighbourhoods satisfy the
latter condition. Since i) and ii) hold for all the worlds in the model, M is a countermodel for the sequent.
Initial sequents and propositional rules - same as NCDL Modal rules G{Γ ⇒ ∆, [⇒ A]i} G{Γ ⇒ ∆, KiA} K R G{A, KiA, Γ ⇒ ∆} G{KiA, Γ ⇒ ∆} K T G{Γ, KiA ⇒ ∆, [KiA, Σ ⇒ Π]i} G{Γ, KiA ⇒ ∆, [Σ ⇒ Π]i} Tr1 G{Γ, KiA, ⇒ ∆, [Σ, KiA ⇒ Π] i} G{Γ ⇒ ∆, [Σ, KiA ⇒ Π]i} Tr2 Fig. 3. Rules of NS5i
5
Relationship with S5
iAs mentioned, the operator Ki can be defined by KiA = ⊥ 4i¬A. If we adopt
this definition, restrict the language to FS5i = p | ⊥ | A → B | KiA, and
apply the rules of NCDLto these formulas (Rem. 1), we obtain a nested sequent
calculus for a multi-agent modal epistemic logic, where the knowledge operator corresponds to the modality. The proof system, called NS5i, captures
multi-agent logic S5i.
Nested sequents of NS5i are interpreted as NCDL nested sequents, with the
difference that NS5i does not need conditional blocks to capture the simpler
se-mantics of S5i. Observe that the rules of NS5i are essentially the multi-agent
versions of the standard nested sequent rules for single-agent S5 [5, 16, 11]. But while the nested sequent structure is an overkill for S5, it is necessary to capture S5i. To the best of our knowledge, the only published sequent calculus for S5i is
Poggiolesi’s hypersequent calculus, which uses syntactic labels for the agents [15]. The connection between mono-agent CDL and S5 is known since [10]: As men-tioned above, counterfactual logic VTA is the mono-agent system corresponding to CDL. But a Kripke-style accessibility relation R can be obtained from (mono-agent) neighbourhood models by setting R(x, y) if and only if y ∈S N (x). For VTA this yields an equivalence relation, thus characterizing modal logic S5. The relation can be used to evaluate formulas KA, i.e., formulas ⊥ 4 ¬A7. For
A ∈ FS5i, define T (A) ∈ FCDL to be the formula obtained by replacing every
occurrence of KiA with ⊥ 4i¬A. The translation is lifted to nested sequents in
the obvious way. By generalizing Lewis’ argument to the multi-agent case, we obtain the following:
Lemma 5. If A is a theorem of S5i, then T (A) is a theorem of CDL.
Completeness of the nested calculus for S5iseems to be unpublished, but
consid-ered folklore in the nested sequent community. Using the previous proposition, it can be obtained proof-theoretically from the completeness of NCDL.
Theorem 3. The calculus NS5i is sound and complete w.r.t. modal logic S5i.
7
Evaluating KA at a world x corresponds to evaluating ⊥ 4 ¬A in the outer neigh-bourhood of N (x). For this reason, Lewis calls S5 the outer modal logic of VTA.
Proof (Sketch). Soundness can be proved directly (standard). For complete-ness, we only sketch the main argument. We claim that for a sequent S = KiA1, . . . , KiAn, Γ ⇒ ∆, KiB1, . . . , KiBm, if there is a derivation of T (S) =
⊥ 4i ¬A1, . . . , ⊥ 4i ¬An, Γ ⇒ ∆, ⊥ 4i ¬B1, . . . , ⊥ 4i ¬Bm in NCDL, then
there is a derivation of the original sequent S in NS5i. If T (S) is derivable in
NCDL, then it must have been derived (modulo rule permutations) either by an
application of T or by multiple applications of R 4, followed by applications of L 4 and com, and finally jump. In the former case, the first premiss of the appli-cation of T contains a block (⊥ Ci⊥) and is derivable via jump, while the right
premiss modulo propositional rules is just the premiss of KT. In the latter case, after (backwards) applications of R 4, we first reach the sequent:
⊥ 4i¬A1, . . . , ⊥ 4i¬An, Γ ⇒ ∆, (⊥ Ci ¬B1), . . . , (⊥ Ci¬Bm).
Similarly to the case of T, the left premiss in any (backwards) application of L 4 to a formula ⊥ 4i ¬A` and a block (⊥ Ci ¬Bk) is derivable, since it contains
the conditional block (⊥ Ci⊥). The other premiss of an application of L 4 is:
⊥ 4i¬A1, . . . , ⊥ 4i ¬An, Γ ⇒ ∆, (Aj, ⊥ Ci ¬B1), . . . , (⊥ Ci¬Bm).
Exhaustive backwards applications of L 4 yield the sequent
⊥ 4i¬A1, . . . , ⊥ 4i¬An, Γ ⇒ ∆, (Σ Ci ¬B1), . . . , (Σ CiBm)
where all blocks have the same Σ = ¬A1, . . . , ¬An. Hence the rule of com is not
really necessary: with applications of L 4 until saturation we obtain the same sequent as with mixed applications of L 4 and com. Finally, by applications of jump and of the rules for negation to the above sequent we reach the sequent
T (S∗) = ⊥ 4i¬A1, . . . , ⊥ 4i¬An, Γ ⇒ ∆, [Σ ⇒ B1]i, . . . , [Σ ⇒ Bm]i.
The corresponding NS5i sequent S
∗ is the same sequent that can be obtained
from Γ, KiA1, . . . , KiAn⇒ ∆, KiB1, . . . , KiBmby applying first rule KR to all
KiB1, . . . , KiBmand then Tr1 exhaustively on KiA1, . . . , KiAn.
Thus, the nested calculus NS5i simulates by macro-steps NCDL derivations
in the restricted language FS5i. Since the structure of conditional blocks is not
needed, the rules of com, Tr3 and Tr4 become superfluous and have no
corre-sponding rules in NS5i. Rule Tr2 simulates rule Tr2. ut
6
Conclusions
We have presented the first internal calculus NCDL for the multi-agent logic of
conditional beliefs CDL. The calculus manipulates nested sequents, where the nesting is determined by nested beliefs of different agents. The calculus provides a decision procedure for the logic. Since CDL contains as a fragment multi-agent S5i, by specialising the rules of NCDL to that fragment we obtain a natural
internal calculus for S5i. CDL logic in itself can be extended to formalise the
dynamics of beliefs induced by different kinds of announcements [1]. We plan to study how to extend our calculus to deal with the dynamic extension of CDL.
References
1. Baltag, A., Smets, S.: Conditional doxastic models: A qualitative approach to dy-namic belief revision. Electronic Notes in Theoretical Computer Science 165, 5–21 (2006)
2. Baltag, A., Smets, S.: A qualitative theory of dynamic interactive belief revision. Logic and the foundations of game and decision theory (LOFT 7) 3, 9–58 (2008) 3. Baltag, A., Smets, S., et al.: The logic of conditional doxastic actions. Texts in
Logic and Games, Special Issue on New Perspectives on Games and Interaction 4, 9–31 (2008)
4. Board, O.: Dynamic interactive epistemology. Games and Economic Behavior 49(1), 49–80 (2004)
5. Br¨unnler, K.: Deep sequent systems for modal logic. Arch. Math. Log. 48, 551–577 (2009)
6. Girlando, M., Lellmann, B., Olivetti, N., Pozzato, G.L.: Standard sequent calculi for Lewis’ logics of counterfactuals. In: Michael, L., Kakas, A. (eds.) Logics in Artificial Intelligence. JELIA 2016., Lecture Notes in Computer Science, vol. 10021, pp. 272–287. Springer, Cham (2016)
7. Girlando, M., Lellmann, B., Olivetti, N., Pozzato, G.L.: Hypersequent calculi for lewis’ conditional logics with uniformity and reflexivity. In: International Confer-ence on Automated Reasoning with Analytic Tableaux and Related Methods. pp. 131–148. Springer (2017)
8. Girlando, M., Negri, S., Olivetti, N., Risch, V.: The logic of conditional beliefs: Neighbourhood semantics and sequent calculus. In: Advances in Modal Logic. pp. 322–341 (2016)
9. Girlando, M., Negri, S., Olivetti, N., Risch, V.: Conditional beliefs: from neigh-bourhood semantics to sequent calculus. The Review of Symbolic Logic pp. 1–44 (2018)
10. Lewis, D.K.: Counterfactuals. Blackwell (1973)
11. Marin, S., Straßburger, L.: Label-free modular systems for classical and intuitionis-tic modal logics. In: Gor´e, R., Kooi, B.P., Kurucz, A. (eds.) AiML 10. pp. 387–406. College (2014)
12. Negri, S.: Proof theory for non-normal modal logics: The neighbourhood formalism and basic results. IFCoLog Journal of Logic and its Applications (2017)
13. Olivetti, N., Pozzato, G.L.: A standard internal calculus for Lewis’ counterfac-tual logics. In: Nivelle, H.D. (ed.) TABLEAUX 2015, Lecture Notes in Artificial Intelligence, vol. 9323, pp. 270–286. Springer International Publishing (2015) 14. Pacuit, E.: Neighbourhood semantics for modal logics. Springer (2017)
15. Poggiolesi, F.: A cut-free simple sequent calculus for modal logic s5. The Review of Symbolic Logic 1(1), 3–15 (2008)
16. Poggiolesi, F.: The method of tree-hypersequents for modal propositional logic. In: Makinson, D., Malinkowski, J., Wansing, H. (eds.) Towards Mathematical Philos-ophy, Trends In Logic, vol. 28, pp. 31–51. Springer Science + Business Media B.V. (2009)
17. Stalnaker, R.: Belief revision in games: forward and backward induction 1. Math-ematical Social Sciences 36(1), 31–56 (1998)
